We start with an overview of a class of submodular functions called SCMMs (sums of concave composed with non-negative modular functions plus a final arbitrary modular). We then define a new class of submodular functions we call deep submodular functions or DSFs. We show that DSFs are a flexible parametric family of submodular functions that share many of the properties and advantages of deep ne...

We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n) (O(n) oracle calls). We therefore propose a fast approximate method to minimize arbitrary submodular functions. For a large ...

We present a provably more efficient implementation of the Minimum Norm Point Algorithm conceived by Fujishige than the one presented in [FUJI06]. The algorithm solves the minimization problem for a class of functions known as submodular. Many important functions, such as minimum cut in the graph, have the so called submodular property [FUJI82]. It is known that the problem can also be efficien...

Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n) and hence rather impractical. In this paper...

This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(nEO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O((nEO+n) log nM) time. The strongly polynomial v...

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to effi...

In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation gu...

Given a finite set V with n elements, a function f : 2 → Z is submodular if for all X,Y ⊆ V , f(X ∪ Y ) + f(X ∩ Y ) ≤ f(X) + f(Y ). Submodular functions frequently arise in combinatorial optimization. For example, the cut function in a weighted undirected graph and the rank function of a matroid are both submodular. Submodular function minimization is the problem of finding the global minimum o...

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [28]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the “curvature” of the submodular function, and provide lower and upper bounds that refine...

Many set functions F in combinatorial optimization satisfy the diminishing returns property F{A U X) — F(A) > F(A' U X) — F(A') for A C A!. Such functions are called submodular. A result from Nemhauser etal. states that the problem of selecting ^-element subsets maximizing a nondecreasing submodular function can be approximated with a constant factor (1 — 1/e) performance guarantee. Khuller eta...